遗憾/惋惜/难过 all can be used in this context, but IMHO none of them is the Chinese counterpart of "sorry".
Using the example in this reference,
I am sorry to hear that your brother passed away.
When it's in Chinese, it seems to imply the speaker knows the person who died and personally feels sad for it.
People also say “我*为你*感到难过” (I feel sad for you), but it's not really common in daily life. You'll probably hear it in televisions and read it in books, but I don't recall a single instance people around me actually say that. It doesn't mean people don't express pity over somebody else's demise, but they say other things like “我能体会你的感受” (I can imaging how bad you feel) etc.